$2\mathrm{.}13*$ Let $S$ be a set of $n$ disjoint triangles in the plane. We want to find a set of $n-1$ segments with the following properties:

Each segment connects a point on the boundary of one triangle to a

point on the boundary of another triangle.

The interiors of the segments are pairwise disjoint and they are disjoint

from the triangles.

Together they connect all triangles to each other, that is$,$ by walking

along the segments and the triangle boundaries it must be possible to

walk from a triangle to any other triangle.

Develop a plane sweep algorithm for this problem that runs in $O(nlogn)$ time. State the events and the data structures that you use explicitly, and describe the cases that arise and the actions required for each of them.

Also state the sweep invariant.